576 CHAPTER 29. FIRST ORDER SCALAR ODE
49. Solve: y′+3cos(t)y = 4(cos t)e−3sin t , y(0) = 1.
50. Solve: y′+ tan(t)y = cos(t) ,y(0) =−2.
51. Solve: x2dy+(4x2− xy+3y2
)dx = 0, y(2) =−2.
52. Solve:( 7
2 y−2x)
dx+(x− 9
4 y)
dy = 0 which contains the point (x,y) = (1,2) .
53. Solve: x2dy+(3x2− xy+2y2
)dx = 0, y(2) =−3.
54. Solve:(x3−6x2y− y3
)dx +
(6x3 + xy2
)dy = 0, y(2) = −3. Graph the integral
curve.
55. Solve: (2y−3x)dx+(2x− 4
3 y)
dy= 0 which contains the point (x,y)= (1,2) . Graphthe integral curve.
56. Solve: y′+5cos(3t)y = 2e−(5/3)sin3t cos3t, y(0) = 2.
57. Solve: x2dy+(5x2− xy+5y2
)dx = 0, y(−2) =−2.
58. Solve:(3x+ 19
4 y)
dx+(−4x− 9
4 y)
dy = 0 which contains the point (x,y) = (1,2) .
59. Solve:(x3−3x2y− y3
)dx+
(3x3 + xy2
)dy = 0, y(3) =−1.
60. Solve: (y)dx+(x+4y)dy = 0 which contains the point (x,y) = (1,2) .
61. Solve: 5(t6)
y+ y′ =−5t6et7, y(1) = 1.
62. Solve: x2dy+(6x2− xy+5y2
)dx = 0, y(3) = 3.
63. Find the solutions to the equation y′+ y(3cos t) = 3(cos t)e−3sin t .
64. Solve: (y−2x)dx+( 9
2 y− x)
dy = 0 which contains the point (x,y) = (1,2) .
65. Solve: x2dy+(2x2− xy+ y2
)dx = 0, y(2) =−1.
66. Find the solutions to the equation y′+2ty = tet2.
67. Solve:( 7
3 y−2x)
dx+(x− 4
3 y)
dy = 0 which contains the point (x,y) = (1,2) .
68. Solve: y′+ tan(2t)y = cos2t,y(0) = 2.
69. Find the general solution to the equation
y′+(4x3 + x2 +3x
)y = exp
(−x4− 1
3x3− 3
2x2)
ln(x+1)
70. Show that the following initial value problem fails to have a unique solution.
y′ = y1/(2n+1),y(0) = 0,n a positive integer.